Nội dung được dịch bởi AI, chỉ mang tính chất tham khảo
Sự tồn tại và hành vi tập trung của các nghiệm trạng thái cơ bản cho một lớp phương trình Schrödinger cận tuyến tổng quát trong ℝN
Tóm tắt
Trong bài báo này, chúng tôi nghiên cứu phương trình Schrödinger cận tuyến tổng quát $$ - \text{div}({\varepsilon ^2}{g^2}(u)\nabla u) + {\varepsilon ^2}g(u)g'(u){\left| {\nabla u} \right|^2} + V(x)u = K(x){\left| u \right|^{p - 2}}u,\;\;\;\;x \in {\mathbb{R}^N},$$ trong đó N ≥ 3, ε > 0, 4 < 22*, g ϵ C1 (ℝ, ℝ+), V ϵ C (ℝN) ∩ L∞ (ℝN) có điểm cực tiểu toàn cục dương, và K ϵ C(ℝN) ∩ L∞ (ℝN) có điểm cực đại toàn cục dương. Bằng cách sử dụng một sự thay đổi biến, chúng tôi xác định được sự tồn tại và hành vi tập trung của các nghiệm trạng thái cơ bản cho vấn đề này và thiết lập một hiện tượng suy giảm theo cấp số nhân.
Từ khóa
#phương trình Schrödinger #nghiệm trạng thái cơ bản #hành vi tập trung #suy giảm theo cấp số nhânTài liệu tham khảo
citation_title=Applied Nonlinear Analysis, Pure and Apllied Mathematics; citation_publication_date=1984; citation_id=CR1; citation_author=J P Aubin; citation_author=I Ekeland; citation_publisher=John Wiley & Sons, Inc
citation_journal_title=J Differential Equations; citation_title=Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity; citation_author=C O Alves, M A S Souto; citation_volume=254; citation_publication_date=2013; citation_pages=1977-1991; citation_id=CR2
citation_journal_title=Phys Rep; citation_title=Nonlinear electromagnetic-spin waves; citation_author=F G Bass, N N Nasanov; citation_volume=189; citation_publication_date=1990; citation_pages=165-223; citation_id=CR3
citation_journal_title=Comm Math Phys; citation_title=Global existence of small solutions to a relativistic nonlinear Schrödinger equation; citation_author=A D Bouard, N Hayashi, J Saut; citation_volume=189; citation_publication_date=1997; citation_pages=73-105; citation_id=CR4
citation_journal_title=Commun Pure Appl Math; citation_title=Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents; citation_author=H Brezis, L Nirenberge; citation_volume=36; citation_publication_date=1983; citation_pages=437-477; citation_id=CR5
citation_journal_title=Commun Pure Appl Anal; citation_title=Existence and concentration of solitary waves for a class of quasilinear Schröodinger equations; citation_author=D Cassani, J M do Ó, A Moameni; citation_volume=9; citation_publication_date=2010; citation_pages=281-306; citation_id=CR6
citation_journal_title=Phys Rev Lett; citation_title=Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma; citation_author=X L Chen, R N Sudan; citation_volume=70; citation_publication_date=1993; citation_pages=2082-2085; citation_id=CR7
citation_journal_title=J Math Anal Appl; citation_title=Positive solution to a class of relativistic nonlinear Schröodinger equation; citation_author=Y Cheng, J Yang; citation_volume=411; citation_publication_date=2014; citation_pages=665-674; citation_id=CR8
citation_journal_title=Appl Math Comput; citation_title=Soliton solutions to a class of relativistic nonlinear Schröodinger equations; citation_author=Y Cheng, J Yang; citation_volume=260; citation_publication_date=2015; citation_pages=342-350; citation_id=CR9
citation_journal_title=Appl Math Lett; citation_title=Non-Nehari manifold method for a class of generalized quasilinear Schröodinger equations; citation_author=J H Chen, X H Tang, B T Cheng; citation_volume=74; citation_publication_date=2017; citation_pages=20-26; citation_id=CR10
citation_journal_title=Mediterr J Math; citation_title=Ground states for a class of generalized quasilinear Schröodinger equations in ℝ; citation_author=J H Chen, X H Tang, B T Cheng; citation_volume=14; citation_publication_date=2017; citation_pages=190; citation_id=CR11
citation_journal_title=J Math Phys; citation_title=Existence and nonexistence of positive solutions for a class of generalized quasilinear Schrodinger equations involving a Kirchhoff-type perturbation with critical Sobolev exponent; citation_author=J H Chen, X H Tang, B T Cheng; citation_volume=59; citation_publication_date=2018; citation_pages=021505; citation_id=CR12
citation_journal_title=J Fixed Point Theory Appl; citation_title=Ground state sign-changing solutions for a class of generalized quasilinear Schrodinger equations with a Kirchhoff-type perturbation; citation_author=J H Chen, X H Tang, B T Cheng; citation_volume=19; citation_publication_date=2017; citation_pages=3127-3149; citation_id=CR13
Chen J H, Tang X H, Cheng B T. Existence and concentration of ground state solutions for a class of generalized quasilinear Schrodinger equation in ℝ2. Preprint
citation_journal_title=J Differ Equ; citation_title=Critical exponents and solitary wave solutions for generalized quasilinear Schrodinger equations; citation_author=Y Deng, S Peng, S Yan; citation_volume=260; citation_publication_date=2016; citation_pages=1228-1262; citation_id=CR15
citation_journal_title=J Differ Equ; citation_title=Positive solition solutions for generalized quasilinear Schröodinger equations with critical growth; citation_author=Y Deng, S Peng, S Yan; citation_volume=258; citation_publication_date=2015; citation_pages=115-147; citation_id=CR16
citation_journal_title=J Math Phys; citation_title=Nodal soliton solutions for generalized quasilinear Schröodinger equations; citation_author=Y Deng, S Peng, J Wang; citation_volume=55; citation_publication_date=2014; citation_pages=051501; citation_id=CR17
citation_journal_title=J Math Phys; citation_title=Nodal soliton solutions for quasilinear Schröodinger equations with critical exponent; citation_author=Y Deng, S Peng, J Wang; citation_volume=54; citation_publication_date=2013; citation_pages=011504; citation_id=CR18
citation_journal_title=Calc Var Partial Differ Equ; citation_title=Solitary waves for a class of quasilinear Schrodinger equtions in dimension two; citation_author=J M do Ó, U Severo; citation_volume=38; citation_publication_date=2010; citation_pages=375-315; citation_id=CR19
citation_journal_title=J Math Phys; citation_title=Existence of solutions for a generalized elliptic problem; citation_author=M F Furtado, E D Silva, M L Silva; citation_volume=58; citation_publication_date=2017; citation_pages=031503; citation_id=CR20
citation_journal_title=Nonlinearity; citation_title=Existence and concentration of positive solutions for quasilinear Schrodinger equations with critical growth; citation_author=X He, A Qian, W Zou; citation_volume=26; citation_publication_date=2013; citation_pages=3137-3168; citation_id=CR21
citation_journal_title=Z Phys B; citation_title=A general method for the solution of nonlinear soliton and kink Schroödinger equations; citation_author=R W Hasse; citation_volume=37; citation_publication_date=1980; citation_pages=83-87; citation_id=CR22
citation_journal_title=J Phys Soc Japan; citation_title=Large-amplitude quasi-solitons in superfluid films; citation_author=S Kurihara; citation_volume=50; citation_publication_date=1981; citation_pages=3262-3267; citation_id=CR23
citation_journal_title=Acta Math Sci; citation_title=Existence of nontrivial solutions for generalized quasilinear Schröodinger equations with critical or supercritical growths; citation_author=Q Li, X Wu; citation_volume=37B; citation_publication_date=2017; citation_pages=1870-1880; citation_id=CR24
citation_journal_title=Comm Partial Differ Equ; citation_title=Nash-Moser methods for the solution of quasilinear Schröodinger equations; citation_author=H Lange, M Poppenberg, H Teismann; citation_volume=24; citation_publication_date=1999; citation_pages=1399-1418; citation_id=CR25
citation_journal_title=J Math Phys; citation_title=Evolution theorem for a class of perturbed envelope soliton solutions; citation_author=E Laedke, K Spatschek, L Stenflo; citation_volume=24; citation_publication_date=1983; citation_pages=2764-2769; citation_id=CR26
citation_journal_title=Proc Amer Math Soc; citation_title=Soliton solutions for quasilinear Schröodinger equations, I; citation_author=J Liu, Z Q Wang; citation_volume=131; citation_publication_date=2003; citation_pages=441-448; citation_id=CR27
citation_journal_title=J Differ Equ; citation_title=Soliton solutions for quasilinear Schrödingerequations, II; citation_author=J Liu, Y Wang, Z Q Wang; citation_volume=187; citation_publication_date=2003; citation_pages=47-493; citation_id=CR28
citation_journal_title=Comm Partial Differ Equ; citation_title=Solutions for quasilinear Schröodinger equations via the Nehari method; citation_author=J Liu, Y Wang, Z Q Wang; citation_volume=29; citation_publication_date=2004; citation_pages=879-901; citation_id=CR29
citation_journal_title=Phys D; citation_title=On instability of excited states of the nonlinear Schröodinger equation; citation_author=S Cuccagna; citation_volume=238; citation_publication_date=2009; citation_pages=38-54; citation_id=CR30
citation_journal_title=J Math Anal Appl; citation_title=Multiple solutions to a class of generalized quasilinear Schroödinger equations with a Kirchhoff-type perturbation; citation_author=F Li, X Zhu, Z Liang; citation_volume=443; citation_publication_date=2016; citation_pages=11-38; citation_id=CR31
citation_journal_title=Math Methods Appl Sci; citation_title=Ground state solutions and geometrically distinct solutions for generalized quasilinear Schröodinger equation; citation_author=Q Li, K Teng, X Wu; citation_volume=40; citation_publication_date=2017; citation_pages=2165-2176; citation_id=CR32
citation_journal_title=Math Methods Appl Sci; citation_title=Multiple solutions for generalized quasilinear Schroödinger equations; citation_author=Q Li, X Wu; citation_volume=40; citation_publication_date=2017; citation_pages=1359-1366; citation_id=CR33
citation_journal_title=J Math Phys; citation_title=Existence, multiplicity, and concentration of solutions for generalized quasilinear Schröodinger equations with critical growth; citation_author=Q Li, X Wu; citation_volume=58; citation_publication_date=2017; citation_pages=041501; citation_id=CR34
citation_journal_title=J Differ Equ; citation_title=Existence of solition solutions for a quasilinear Schröodinger equation involving critical exponent in ℝ; citation_author=A Moameni; citation_volume=229; citation_publication_date=2006; citation_pages=570-587; citation_id=CR35
citation_journal_title=Phys Rep; citation_title=Nonlinear effects in quasi-one-dimensional models and condensed matter theory; citation_author=V G Makhankov, V K Fedyanin; citation_volume=104; citation_publication_date=1984; citation_pages=1-86; citation_id=CR36
citation_journal_title=Calc Var Partial Differ Equ; citation_title=On the existence of soliton solutons to quasilinear Schröodinger equations; citation_author=M Poppenberg, K Schmitt, Z Q Wang; citation_volume=14; citation_publication_date=2002; citation_pages=329-344; citation_id=CR37
citation_journal_title=Calc Var Partial Differ Equ; citation_title=Local mountain passes for semilinear elliptic problems in unbounded domains; citation_author=M D Pino, P L Felmer; citation_volume=4; citation_publication_date=1996; citation_pages=121-137; citation_id=CR38
citation_journal_title=Phys Rev E; citation_title=Relativistic self-focusing and channel formation in laser-plasma interaction; citation_author=B Ritchie; citation_volume=50; citation_publication_date=1994; citation_pages=687-689; citation_id=CR39
citation_journal_title=Z Angew Math Phys; citation_title=On a class of nonlinear Schröodinger equations; citation_author=P H Rabinowitz; citation_volume=43; citation_publication_date=1992; citation_pages=270-291; citation_id=CR40
citation_journal_title=Nonlinearity; citation_title=Existence of ground states for a modified nonlinear Schrodinger equation; citation_author=D Ruiz, G Siciliano; citation_volume=23; citation_publication_date=2010; citation_pages=1221-1233; citation_id=CR41
citation_journal_title=Calc Var Partial Differ Equ; citation_title=Quasilinear asymptotically periodic Schröodinger equations with critical growth; citation_author=E A Silva, G F Vieira; citation_volume=39; citation_publication_date=2010; citation_pages=1-33; citation_id=CR42
citation_journal_title=J Math Anal Appl; citation_title=Existence and concentration behavior of positive solutions for a quasilinear Schröodinger equation; citation_author=X Shang, J Zhang; citation_volume=414; citation_publication_date=2014; citation_pages=334-356; citation_id=CR43
citation_journal_title=Complex Var Elliptic Equ; citation_title=Standing waves for a class of quasilinear Schroödinger equations; citation_author=Y Shen, Y Wang; citation_volume=61; citation_publication_date=2016; citation_pages=817-84; citation_id=CR44
citation_journal_title=Appl Math Lett; citation_title=Two types of quasilinear elliptic equations with degenerate coerciveness and slightly superlinear growth; citation_author=Y Shen, Y Wang; citation_volume=47; citation_publication_date=2015; citation_pages=21-25; citation_id=CR45
citation_journal_title=Nonlinear Anal: Theory Methods Appl; citation_title=Soliton solutions for generalized quasilinear Schröodinger equations; citation_author=Y Shen, Y Wang; citation_volume=80; citation_publication_date=2013; citation_pages=194-201; citation_id=CR46
citation_journal_title=J Math Anal Appl; citation_title=Existence and concentration of ground state solutions for a subcubic quasilinear problem via Pohožaev manifold; citation_author=W Wang, X Yang, F K Zhao; citation_volume=424; citation_publication_date=2015; citation_pages=1471-1490; citation_id=CR47
citation_journal_title=Nonlinear Differ Equ Appl; citation_title=Bound states to critical quasilinear Schrödinger equations; citation_author=Y Wang, W Zou; citation_volume=19; citation_publication_date=2012; citation_pages=19-47; citation_id=CR48
citation_title=Minimax Theorems; citation_publication_date=1996; citation_id=CR49; citation_author=M Willem; citation_publisher=Birkhöauser
citation_journal_title=Calc Var Partial Differ Equ; citation_title=The concentration behavior of ground state solutions for a fractional Schröodinger-Poisson system; citation_author=Y Yu, F K Zhao, L Zhao; citation_volume=56; citation_publication_date=2017; citation_pages=116; citation_id=CR50
citation_journal_title=J Math Phys; citation_title=Existence of ground state solutions to a generalized quasilinear Schrödinger-Maxwell system; citation_author=X Zhu, F Li, Z Liang; citation_volume=57; citation_publication_date=2016; citation_pages=101505; citation_id=CR51